Quite a few mathematical constants are known to arise in several branches of mathematics (more here). I have no doubt that they are useful to the understanding of some mathematical structures and add to the whole body of knowledge of mathematics as a whole. Sometimes, however, I find myself looking at these constants and I feel unsatisfied. I suspect that more constants can be expressed in more 'fundamental' constants like $\pi, e$ and $\gamma$.
Although Euler probably came close to be a living one, mathematicians of the past did not have computers. We can now use computers to determine te value of the afore mentioned constants with great precision. We can also try to conjecture the exact value of these constants by finding the value of some arithmetic combination of $\pi, e$ and $\gamma$ and the real numbers that corresponds to one of the 'unevaluated' constants with high precision. My question is: have experimental mathematicians 'found' the value(s) of some of the constants I described (by the the method I described or a similar one)?
P.S. If someone knows of a reference to a paper/book that summarizes some results in this subield of a field, I would be grateful to him/her.